Radial symmetry

The designer should attempt to keep the eapaeitors radially symmetrie from the ripple eurrent souree for both sides of the loop. 

Note that pressure is treated as a function of z alone. This is consistent with the assumption of negligible Vr- Equation (8.63) is subject to the boundary conditions of radial symmetry, dVJdr = 0 at r = 0, and zero slip at the wall, Fz = 0 SLtr = R. 

We turn now to the numerical solution of Equations (9.1) and (9.3). The solutions are necessarily simultaneous. Equation (9.1) is not needed for an isothermal reactor since, with a flat velocity profile and in the absence of a temperature profile, radial gradients in concentration do not arise and the model is equivalent to piston flow. Unmixed feed streams are an exception to this statement. By writing versions of Equation (9.1) for each component, we can model reactors with unmixed feed provided radial symmetry is preserved. Problem 9.1 describes a situation where this is possible. 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

This formulation assumes radial symmetry in the polishing process. With the substitution 6 = 2(p and cos 6=1 — 2 sin q>, the effective radial velocity... 

Visual deflnitions of flow patterns in vertical flow appear to cause more diflBculty than do those in horizontal flow. As the gas rate increases at a constant liquid rate, a dispersed type of flow will be reached at lower gas velocities in a vertical tube than in a horizontal one because of the influence of gravity in causing back flow of liquid. Also, vertical flow patterns tend toward radial symmetry, which is not the case in horizontal flow. A classification of vertical flow patterns based largely on air-water mixtures is given below. Surprisingly little work has been carried out for the vertical upward flow of components other than these, or steam-water mixtures. 

Some detailed analyses of the mechanics of vertical slug flow (N4, G8) and vertical annular flow (A2, Cl, C6, LI) have appeared in recent years, and give a clearer picture of the microscopic behavior in these patterns. In general, this type of analysis is yielding better results with many vertical flow patterns because of their radial symmetry, than with horizontal flow patterns. 

TrochophorE A form of free swimming larva characteristic of several phyla of aquatic invertebrates typically possessing a radial symmetry, with a ring of cilia around it, an eye-spot, and a digestive tract with mouth and anal openings. 

Due to the radial symmetry of the three configurations, a 2D model is employed. Figure 4.20 depicts the geometry considered for case 1, i.e. current collectors at the tube ends (not in scale). In cases 2, an additional 16 pieces of conductive material (nickel), with a square section of 1.5 x 1.5 mm, are considered on the cathode (external surface of the tube). In case 3 the conductive materials are present both inside and outside the tube. It should be noted that in case 2 and 3, cylindrical symmetry is not obvious anymore. However, it is reasonable to assume that voltage losses are mainly proportional to ohmic in-plane losses, i.e. to the length of the current path along the tube. In case 2 and 3, this distance is constant in each longitudinal tube cross-section, therefore a cylindrical symmetry is assumed. 

The polar form of the name of this monomer in this particular case is nearly the same as the rectangular form however, it has the advantage that it emphasizes the radial symmetry associated with dendimers viz. ... 

In the last twenty years, in conclusion, paleontology has discovered that in Cambrian times there have been not one but three different explosions ofanimallife one documented by trace fossils, a second which left behind small shelly fossils, and finally the classical explosion that was dominated by trilobites. It must also be added that Cambrian life was preceded by the so-called Ediacara fauna, a vast assembly of soft-bodied animals (almost all with radial symmetry). Many scholars now regard them as a failed evolutionary experiment, while others believe that they may have left modified descendants (Figure 7.1). 

Find the equilibrium equations for a system with radial symmetry (Fig. P4.7.1). 

To start with, let us determine the stress and the deformation of a hollow sphere (outer radius J 2, inner radius R ) under a sudden increase in internal pressure if the material is elastic in compression but a standard solid (spring in series with a Kelvin-Voigt element) in shear (Fig. 16.1). As a consequence of the radial symmetry of the problem, spherical coordinates with the origin in the center of the sphere will be used. The displacement, obviously radial, is a function of r alone as a consequence of the fact that the components of the strain and stress tensors are also dependent only on r. As a consequence, the Navier equations, Eq. (4.108), predict that rot u = 0. Hence, grad div u = 0. This implies that... 

Owing to the radial symmetry of the problem, and taking into account that the excitation is harmonic, the solution for the differential equation can be assumed to be... 

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